Implementation of the GMRES method using householder transformations
SIAM Journal on Scientific and Statistical Computing - Telecommunication Programs at U.S. Universities
SIAM Journal on Numerical Analysis
A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds
Journal of Computational Physics
Journal of Computational Physics
On the Fourier Extension of Nonperiodic Functions
SIAM Journal on Numerical Analysis
Journal of Computational Physics
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We present fast, spatially dispersionless and unconditionally stable high-order solvers for partial differential equations (PDEs) with variable coefficients in general smooth domains. Our solvers, which are based on (i) A certain "Fourier continuation" (FC) method for the resolution of the Gibbs phenomenon on equi-spaced Cartesian grids, together with (ii) A new, preconditioned, FC-based solver for two-point boundary value problems for variable-coefficient Ordinary Differential Equations, and (iii) An Alternating Direction strategy, generalize significantly a class of FC-based solvers introduced recently for constant-coefficient PDEs. The present algorithms, which are applicable, with high-order accuracy, to variable-coefficient elliptic, parabolic and hyperbolic PDEs in general domains with smooth boundaries, are unconditionally stable, do not suffer from spatial numerical dispersion, and they run at Fast Fourier Transform speeds. The accuracy, efficiency and overall capabilities of our methods are demonstrated by means of applications to challenging problems of diffusion and wave propagation in heterogeneous media.